It appears that you are asking which groups $G$ occur as quotients of the fundamental group $A_g=\pi_1(\Pi_g)$ for some $g$. Then the answer is "all finitely generated $G$", where $g$ can be taken to be the rank of $G$, i.e. the least cardinality of a generating set. If $G$ is a finite simple group, then it is known to be generated by 2 elements, so we can take $g=2$.

Proof Using the standard one-relator presentation of $A_g$ as $$\langle a_1,\ldots,a_g,b_1,\ldots,b_g\,|\,[a_1,b_1]\cdots [a_g,b_g]\rangle,$$ the map sending $a_i, 1\leq i\leq g$ to some generators of $G$ and each $b_i$ to the identity of $G$ uniquely extends to a group surjection $A_g\to G$.

It appears that you are asking which groups $G$ occur as quotients of the fundamental group $\Pi_g=\pi_1(\Sigma_g)$ for some $g$. Then the answer is "all finitely generated $G$", where $g$ can be taken to be the rank of $G$, i.e. the least cardinality of a generating set. If $G$ is a finite simple group, then it is known to be generated by 2 elements, so we can take $g=2$.

Proof Using the standard one-relator presentation of $\Pi_g$ as $$\langle a_1,\ldots,a_g,b_1,\ldots,b_g\,|\,[a_1,b_1]\cdots [a_g,b_g]\rangle,$$ the map sending $a_i, 1\leq i\leq g\,$ to some generators of $G$ and each $b_i$ to the identity of $G$ uniquely extends to a group surjection $\Pi_g\to G$.